The asymptotic cluster-weight conjecture for H-coloring tori

Let $H$ be a graph, let $\Lambda$ be a vector of vertex weights, and let $mgeq 2$ be even. Write $Hom({\mathbb Z}^d_m,H)$ for the set of $H$-colorings of the $m$-ary $d$-dimensional torus, let $\mathcal M_\Lambda(H)$ be the set of dominant pairs, and let $C_\Lambda(A,B)$, $\eta_\Lambda(H)$, and $L_\Lambda(A,B,d)$ have the meanings introduced above. The asymptotic cluster-weight conjecture. For all $H$, $\Lambda$, and even $m\geq 2$, there is a decomposition of $Hom({\mathbb Z}^d_m,H)$ satisfying the conditions of the structural theorem and, for every $(A,B)\in\mathcal M_\Lambda(H)$, satisfying

$$w_\Lambda(C_\Lambda(A,B))=\eta_\Lambda(H)^{m^d/2}\exp\left\{m^dL_\Lambda(A,B,d)(1+o(1))\right\}$$

as $d\to\infty$. The conjecture is known for the hard-core model on $Q_d$ with all positive activities, and for proper $3$-colorings of $Q_d$, but is open in general.